AND

NASA TECHNICAL NOTE NASA TN D-7866

-J__ U.S.Nco

(NASA-TN-D-7866) THE ANNULAR MOMENTUMCONTROL DEVICE (AMCD) (NASA) 38 p HC $3.75 N75-18302

CSCL 22B

Unclas. .H1/18 13553

THE ANNULAR MOMENTUM CONTROL DEVICE (AMCD)

AND POTENTIAL APPLICATIONS

Willard W. Anderson and Nelson J. Groom

Langley Research Center

Hampton, Va. 23665 o .

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. MARCH 1975NATIONAL AERONAUTICS AND SPACE ADMINISTRATION * WASHINGTON, D. C. * MARCH 1975

https://ntrs.nasa.gov/search.jsp?R=19750010230 2018-06-24T03:07:56+00:00Z

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TN D-78664. Title and Subtitle 5. Report Date

THE ANNULAR MOMENTUM CONTROL DEVICE (AMCD) AND March 1975

POTENTIAL APPLICATIONS 6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.

Willard W. Anderson and Nelson J. Groom L-9901

10. Work Unit No.9. Performing Organization Name and Address 506 -19-13-01

NASA Langley Research Center11. Contract or Grant No.11. Contract or Grant No.

Hampton, Va. 23665

13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address Technical NoteNational Aeronautics and Space. Administration

14. Sponsoring Agency CodeWashington, D.C. 20546

15. Supplementary Notes

16. Abstract

An annular momentum control device (AMCD) consisting principally of a spinning rim, a

set of noncontacting magnetic bearings for supporting the rim, a noncontacting electric motor

for driving the rim, and, for some applications, one or more gimbals is described. The deviceis intended for applications where requirements for control torque and momentum storage exist.Hardware requirements and potential unit configurations are discussed. Theoretical considera-

tions for the "passive" use of the device are discussed. Potential applications of the device inother than passive configurations for the attitude control, stabilization, and maneuvering of

spacecraft are reported.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Spacecraft stabilization Unclassified - Unlimited

Spacecraft control actuator

Momentum storage device

Control moment gyroscope STAR Category 3119. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price*

Unclassified Unclassified 36 $3.75

For sale by the National Technical Information Service, Springfield, Virginia 22151

THE ANNULAR MOMENTUM CONTROL DEVICE (AMCD)

AND POTENTIAL APPLICATIONS

By Willard W. Anderson and Nelson J. Groom

Langley Research Center

SUMMARY

An annular momentum control device (AMCD) consisting principally of a spinningrim, a set of noncontacting magnetic bearings for supporting the rim, a noncontacting

electric motor for driving the rim, and, for some applications, one or more gimbals isdescribed. The device is intended for applications where requirements for control torqueand momentum storage exist. Hardware requirements and potential unit configurationsare discussed. Theoretical considerations for the "passive" use of the device are dis-

cussed. Potential applications of the device in other than passive configurations for theattitude control, stabilization, and maneuvering of spacecraft are reported.

INTRODUCTION

The use of stored angular momentum for the purpose of controlling the attitude of

fine pointing or long duration spacecraft, where either environment contamination orexcessive fuel use prohibit reaction jet usage, has become almost universal. Applicationsof the concept include spinning spacecraft (Tiros), dual-spin spacecraft (OSO), momentumwheel stabilized spacecraft (ITOS), reaction-wheel stabilized spacecraft (OAO), and,recently, the control moment gyroscope system used for stabilizing Skylab.

Each of these methods has advantages and disadvantages. Spinning the spacecraft(or a portion of it) to achieve attitude stability is simple and reliable, yet the spacecraftitself cannot be utilized fully because of its rotation. Also, any momentum axis reorien-tation maneuvers require external torques for momentum precession and the artificialgravity field produced may run counter to payload requirements for zero gravity. Sta-bilizing the spacecraft by utilizing a momentum wheel which provides gyroscopic stiffnessequivalent to spinning the vehicle itself allows a nonspinning spacecraft and permits arbi-trary orientations about the roll axis for the purpose of pointing onboard experiments.This technique does not overcome the inability to reorient or maneuver the spacecraftabout all three axes without external torques since the spin axis of the momentum wheelis fixed with respect to the spacecraft. The use of three reaction wheels with axes alinedwith the spacecraft axes allows complete spacecraft active attitude control. However,

reaction-wheel momentum must be limited to relatively low amounts because of an

excessive requirement for power when directly producing a torque on a rapidly spinning

flywheel. The limitation on reaction-wheel momentum can be overcome by using a con-

trol moment gyro (CMG) system, which uses constant speed wheels and develops preces-

sion torques through controlled slewing of gimbaled flywheels. However, to achieve the

smooth low-level torques necessary for fine pointing requires precise control of very

low gimbal slew rates. These low gimbal rates are inherently limited by the requirement

for high servo stiffness and thus high friction torque, unless extreme mechanical preci-

sion and resultant high cost are involved.

The purpose of this paper is to introduce a new development in the field of momen-

tum storage devices, the annular momentum control device (AMCD). 1 The spinning

assembly of the device is discussed in detail and its advantages relative to conventional

flywheel design are presented. The use of the spin assembly as a momentum wheel

("passive" case) with spin axis control for spacecraft with orientation requirements such

as those of Tiros, ITOS, or TACSAT is described in detail.

The use of the device for active pointing control, maneuvering, and other applica-

tions of spacecraft and experiment control is discussed in general terms. Emphasis is

placed on the use of the device in applications where a large radius is allowable and ben-

eficial since those applications are new. ' The device can be used as the spin assembly

for any of the conventional momentum storage devices including reaction wheels and con-

trol moment gyros.

SYMBOLS

Values are given in both SI and U.S. Customary Units. The measurements and

calculations were made in U.S. Customary Units.

A rim cross-sectional area

E transformation matrix

F magnetic bearing forces

G external torques

H angular momentum

'Patent pending for "Annular Momentum Control Device," NASA CaseNo. LAR 11051-1.

2

Ha total angular momentum of AMCD

Hs spacecraft total angular momentum

I moment of inertia

Ia transverse AMCD rim inertia

Is transverse spacecraft inertia

Im() imaginary part of complex root

Kk magnetic bearing linear displacement gain

Kj magnetic bearing linear rate gain

Ks flywheel energy shape factor (see eq. (Al))

Ks flywheel momentum shape factor (see eq. (1))

K k magnetic bearing angular displacement gain

Kk magnetic bearing angular rate gain

k radius of gyration

m mass

Re() real part of complex root

r radius

s distance along rim; also, Laplace operator

T interaction torque

TR flywheel energy

t time

v arbitrary vector

X,Y,Z coordinate axes

Y Young's modulus

a telescope azimuth

0 telescope elevation

y angle between axis of rotation and X-axis (see eqs. (C14))

6 spacecraft rotation angle

eH spin momentum ratio, Hs/Ha

EI transverse inertia ratio, Ia/I s

7 rim vibration amplitude

9 transformation angle, Y-axis rotation angle

rim rotation angle about axis of rotation (see eqs. (C14))

rim rotation angle

p mass density

(Psyst)max maximum system damping (see eq. (12))

P.0 damping coefficient

a working stress

r magnetic bearing gain ratio, Kk/K k

X-axis rotation

4

Ilp uncoupled rim precessional frequency

W angular velocity

' angular velocity cross product matrix

Wf rim fundamental frequency

WH approximate precession frequency, (Ha + Hs)/(Ia + Is)

oo stiffness factor, (KX/Is)1/2

Subscripts:

a AMCD rim

b body

d damped

f body fixed

i inertial

s spacecraft

T total

x,y,z coordinates

Superscripts:

c coordinate axes

-1 matrix inversion

* nondimensional with respect to wH

Dots over symbols denote derivatives with respect to time.

5

AMCD SPIN ASSEMBLY DESCRIPTION

The spin assemblies of momentum wheels, reaction wheels, and control moment

gyroscopes used in attitude control systems for spacecraft to date have been of nearly

conventional terrestrial design (shaft-driven steel flywheels with ball bearings). The

AMCD spin assembly configuration is based on space usage (vacuum and zero gravity)

and consists of a rotating rim (no central hub) suspended by noncontacting magnetic bear-

ings and powered by a noncontacting linear electromagnetic motor. (See fig. 1.)

Magnetic bearings

Rim

Rim drive motor

Figure 1.- Basic AMCD configuration.

The choice of configuration is based on several factors. The first factor is a

desire to obtain maximum angular momentum for a given flywheel mass. The relation-

ship of these two variables is discussed in appendix A (eq. (A3)) and is given as follows:

H= k(2K s a 1 / 2

where H is the angular momentum, m the mass, k the radius of gyration, Ks a

dimensionless shape factor, a the allowable working stress, and p the density of the

flywheel. A consideration when optimizing H/m is the specific stress o/p of the

material. From reference 1, working values of the square root of specific stress for

6

conventional metals (steels and aluminums) lie below 350 m/sec (1150 ft/sec), the

stronger maraging steels approaching 500 m/sec (1640 ft/sec). Values of specific stressfor the filaments of the stronger composite materials range from 1060 m/sec (3480 ft/sec)

for boron and graphite to 1230 m/sec (4050 ft/sec) for the Dupont material PRD-49

(Kevlar). Clearly, from the aspect of specific stress, the composite materials are better.However, to achieve full utilization of the material strength requires a unidirectional

layup of the composite.

Another consideration is the combined effect of size (radius of gyration) and shape.

For a flywheel of constant shape, H/m is proportional to the radius of gyration k, and

it is clear, therefore, that the largest possible k is optimum. However, flywheel size

is normally limited by outside diameter and not by the radius of gyration. Equation (A3)

is thus rewritten as

= ro(2Ks 1/2 (1)

where r o is the outside radius of the flywheel and

Ks = Ksro2

The maximum value for K' for the nine flywheel shapes considered in reference 2is K' = 0.500 for the thin rim. It is interesting to note that not only is the thin rim an

optimum flywheel shape but it is also one of the few flywheel shapes that will allow full

utilization of the filament strengths of composite materials by allowing a unidirectional

layup. This fact also makes the AMCD an attractive energy storage device.

The second factor behind the selection of the AMCD configuration is the multiple

problems of wear, reliability, and vacuum lubrication of the ball bearings used in current

and planned control moment gyros. To achieve the large momentum needed for Skylab

and for planned shuttle and shuttle-launched missions requires flywheel speeds at values

where ball bearing design is an empirical art necessitating extensive preflight testing ofthe bearings and lubrication system. As has been evidenced on Skylab, where two out of

three of the control moment gyros developed bearing problems (one causing complete

failure of the unit), poor performance can still result from this empirical design process.

The use of magnetic bearings to support the rim eliminates all wear, requires no lubri-

cation, and increases the reliability to that of the solid-state electronic bearing circuits.

A third factor is the isolation of the rotating rim from the spacecraft afforded by

the magnetic bearings. Spacecraft vibration resulting from reaction wheel or CMG rotor

unbalance and ball bearing element rotation can have a detrimental effect on the fine

pointing of vehicles with high accuracy requirements such as the planned large space

telescope. (See ref. 3.) Although imperfect balance of an AMCD rim would cause rim

7

motion, for the case of all active nonpermanent magnet bearings the motion itself would

produce no disturbance on the spacecraft unless detected by the bearing sensors and then

acted upon by the bearing force coils. For this reason, appropriate notch filtering at the

spin frequency and at the fundamental vibrational frequency and possibly at the higher

harmonics of both should eliminate vibrational input to the vehicle. It is interesting to

note that for the thin-rim AMCD, it is not possible to have resonance between spin and

spin plane vibration frequencies because the rim vibration frequencies, for the assump-

tions of appendix B, are always higher by a factor of at least two. Although the assump-

tions of appendix B do not cover all potential AMCD rim geometries, the equations illus-

trate the stiffening which occurs because of the tangential stress generated by rim

rotation. Also this stiffening has the effect of circularizing the rim and improving bal-

ance at design speed.

A final factor to consider is the ability to command output torques for control

usage directly with no mechanical or electrical breakout torque. This ability is implicit

when the device is used either in the "passive" mode to be discussed later where control

torques are generated by a damped centering of the rim between bearing elements or by

direct command from spacecraft serisors when used in the "active" mode which is also

discussed later.

AMCD SPIN ASSEMBLY HARDWARE CONSIDERATIONS

Because of its unique configuration, the AMCD presents several design constraints

which interact with all components and result in basic requirements for any AMCD

design. Obviously, any number of variations in the design of rim, motor, and/or bearings

could be made in an attempt to better meet the specific design requirements for a given

mission, but some general considerations warrant discussion.

Magnetic Bearing Considerations

For AMCD designs of large radial dimension, a very large weight penalty could

occur if a continuous magnetic bearing around the rim was provided. In practical terms

the total magnetic bearing structure weight could be two to three times that of the rim.

This condition could compromise any system weight advantage presented by the AMCD

design. The practical solution to the bearing problem is to use segmented bearings as

shown in figure 1. The minimum number of bearing segments would be three with a

variable number of elements per segment depending on particular design requirements.

Unfortunately, the use of segmented bearings to solve the bearing weight problem

produces another potentially serious problem. Since the bearings are now in discrete

sections, the rotating rim will see a discontinuous magnetic field. Unless proper choice

8

of rim design is made, this could result in appreciable eddy current and hysteresis

power losses in the rim portion of the bearing magnetic circuit.

Rim Considerations

As mentioned previously, the use of segmented magnetic bearings introduces the

design requirement that the rim portion of the bearing magnetic circuit have low eddy

current and hysteresis losses. Thus, a low reluctance material with a very high resis-

tivity and a narrow hysteresis loop is required. A material that has these characteris-

tics in addition to the structural characteristics required for the rim structure does not

currently exist. A practical solution to the conflicting rim material problem is to use a

high-strength nonmagnetic and nonconducting composite material such as graphite epoxy

for the rim structure and embed a low-loss magnetic material such as a ferrite material

in the main rim structure to serve as the rim portion of the bearing magnetic circuit.

Rim Drive Motor Considerations

As with the magnetic bearings, the weight penalty for a motor stator that is contin-

uous around the rim would be too high. Use of a linear motor employing one or more

stator segments is one obvious solution. Again the problem of conflicting requirements

for the rim material arises. As in the magnetic bearing solution, the rotor part of the

motor magnetic circuit can be embedded in the main rim structure. For example, a

permanent magnet dc brushless drive motor could be designed by utilizing permanent

magnets embedded in the rim with a magnetic field detector to sense the position of a

given magnet for commutation purposes.

AMCD Laboratory Model

An AMCD laboratory model embodying the design philosophy presented is being

built for the Langley Research Center by Ball Brothers Research Corporation. The rim

(fig. 2) is made of a graphite-epoxy composite, is 1.7 meters (5.5 feet) in diameter,weighs 222.4 N (50 pounds), and will rotate at 3000 rpm. The suspension system con-

sists of three segments spaced equidistantly around the rim. Magnetic bearing elements

at each segment interact with low-loss ferrite material embedded in the rim. The rim

will be spun up by a series of stationary electromagnets that both push and pull on small

samarium-cobalt permanent magnets also embedded in the rim.

THEORETICAL CONSIDERATIONS OF A "PASSIVE"

AMCD-SPACECRAFT SYSTEM

The use of the AMCD as a "passive" device involves rigidly mounting the magneticbearings and spin motor to the spacecraft and providing an active spacecraft control loop

9

L-74-8547

Figure 2.- Graphite-epoxy rim for the AMCD

laboratory model.

using spin motor torques for spacecraft roll or spin control. (See fig. 3.) The word

passive is used because spacecraft sensors are not required for stabilization of space-

craft pitch and yaw axes although rim position sensors are required for the active mag-

netic bearings. Nonlinear and linearized equations of motion for the passive AMCD-

spacecraft case are derived in appendix C. These equations were used to generate time

histories and root locus plots which are discussed later. Before presenting these data,

the rotational stability of the device is discussed.

Basic Rotational Stability Considerations

The linear model derived in appendix C does not contain spin axis torques when the

AMCD spin motor torque is controlled to counteract AMCD rim friction (hysteresis and

eddy current losses) since both the AMCD rim and the spacecraft are assumed to be sym-

metrical about the spin axis. This means that there is no mechanism within the model to

allow the coning angle to increase since any increase in the coning angle can only be the

result of a decrease in spin momentum if there are no external torques present.

10

Y

Magnetic bearing Space vehicleelements

1 Spin motor elements

/X

AMCD momentum,H ///

/

/

SZ I / Rotating rimI !I,

Figure 3.- "Passive" AMCD-spacecraft system.

To explain this further, assume that the relative transverse rotation between rim

and spacecraft is small (as has been assumed throughout the report). Thus

HT cos 9 = Iazwaz + Iszwsz (2)

where HT is the total angular momentum of the system, 9 is the angle between theZ-axis and the momentum vector, Iaz and Isz are the rim and spacecraft spin-axismoments of inertia, and waz and wsz are the spin rates of the rim and spacecraft.Differentiating with respect to time yields

=- Iaz' az +Iszcsz (3)HT sin 8

and thus an increase in the cone angle 0 is seen to require a net decrease in spinmomentum. Obviously, if there are no spin axis torques acting directly on the two bodiesand the two bodies are symmetrical about the spin axis (no spin-axis cross-couplingtorques), the spin momentum is constant and therefore 8 is constant.

In any real physical system, internal sources of energy dissipation do exist which

can change the spin momentum. There are many references in the literature dealing withthe subject of rotational stability based on models of dual-spin vehicles containing internalsources of energy dissipation which affect spin momentum. Vernon Landon (ref. 4) is

credited with being the first investigator to explain the stability of dual-spin vehicles whenthese vehicles are not spinning about an axis of maximum inertia. For the purposes ofthis paper, it is assumed that the energy dissipation sources exist almost entirely within

11

the vehicle rather than within the AMCD rim. For this reason the requirement for rota-

tional stability (for the case of small relative transverse rotations) reduces to a require-

ment that the rate of precession of the spin axis (Z-axis) about the total momentum vector

be greater than the rate of spin of the spacecraft. (See ref. 5 for a mathematical deriva-

tion of this requirement.) For the passive AMCD-spacecraft system, this condition

represents a minimum requirement for AMCD rim momentum which must be met to

insure asymptotic rotational stabilities.

The linear model does contain terms which can render the system unstable. How-

ever, the instability would not affect the system cone angle, but rather would allow the

relative transverse rotation between rim and spacecraft to increase in an unstable man-

ner. These instabilities are examined after the characteristic equation for the linear

model is discussed.

Characteristic Equation for the Linearized Model

The nonlinear and linearized equations of motion for the passive AMCD-spacecraft

case are derived in appendix C. The nonlinear equations were used to generate time

histories which are discussed later. The linearized set of equations (eqs. (C21)) has

the following characteristic equation which will be used to discuss system behavior:

s2 E12)s6 + 2Tw,2q I(1 + Is5 + [o2Ei(1 + E) + 2 w 4 1 + 2+ + H2 2 1 + e)H22I 2 s4

+ 2TWoo4(1 +EI)2 + 2Tw2 wH2( +I2 RH 1 H I + + EH ES3 + 4(1 + Ei 2

++ EH I +] E.,/ I)]+ 72 w, 4 H2 (1 + Ei)2 1 + +H) 2j+ (EHWH2 2 + 2 H2 2 + H s2

+ [2Tw 4wH2(1 + Ei) 2 + 2Two 2WH4REH2 + 4H2(1 + E )2 [1 + (T =) 0 (4)

where

T Kk I = IaK I

o2 _ KX Hs

I _ s H = (5)

wH Ha + Hs R I a + I s

Ia + Is Isz

12

OUORINIA PAGE IS

OF'POOP- Q13ALM

and where KX and Kk are the resultant angular and angular rate magnetic bearing

constants, Ia and I s are the AMCD rim and spacecraft transverse moments of inertia,

and Ha and Hs are the AMCD rim and spacecraft spin momentums. (See appendix C.)

The coefficients of equation (4) were used to form a symbolic Routh stability array

by means of a computerized symbolic manipulation routine called MACSYMA. An attempt

to derive simplified conditions for system stability was not successful because of algebraic

complexity. The stability of the system was then examined by using the method of refer-

ence 6. This examination yielded the following two conditions for stability for positive

damping T:

REH <1

1 +EH

o- 2 +1 +E 1 I 2RH +R 1 1- R)( - RRH 1 - EH2 R 2 I + >) 0

WH + EH 2 L+ E H 1 + EH(1 - R) + H 1 + E)2

These conditions agree numerically with the Routh array conditions.

It is informative to discuss simplified versions of equation (4) initially. For the

case where the AMCD rim transverse inertia Ia and the spacecraft spin momentum Hs

are relatively small (e, and EH - 0), equation (4) simplifies to

s2[(T2w4 + wH2) s4 + (2TWoo4 + 2Tcoo2 wH2)s3 + (w,4 + T2wO 4 wH 2 + 2Wom2wH2)52

+ (2TCoo4WH2)s + (WOO4wH2)] = 0 (6)

Referring to the first of equations (5), the variable 7 is seen as a ratio of mag-

netic bearing gains or the required lead on the magnetic bearing rim displacement sig-

nal. (Also see eqs. (C17).) This variable defines system damping. Insight into the

variable woo can be gained by letting the AMCD momentum become large (wH - ).

Equation (6) is reduced to

s2 92 + To 29 2 2=0 7

s2(s2 + TCO 02 S + 2) = 0 (7)

The combined behavior described by this equation is that of uncoupled X- and Y-axis

vibration of the spacecraft (the two complex conjugate roots) with damped frequency

woo, d = woo(1 - P 2) 1 / 2 Po0 < 1)

and damping coefficient

T cooPoo- 2

13

and precession of the system (the two s = 0 roots) about the X- and Y-axes caused byexternal torques about the Y- and X-axis, respectively. Since the momentum is largefor this case, the rate of precession (which is inversely proportional to momentum)would be zero in the limit as wH approached infinity.

The variable WH can be defined by allowing the control gain ratio or lead 7 toapproach infinity. Under this circumstance the AMCD rim is not allowed transversemotion with respect to the spacecraft; this situation is thus likened to a nonflexible dual-spin spacecraft. Equation (4) reduces to

s2(s2 + 2) (s2 + wH2) = 0 (8)

Thus, the variable wH is the frequency of the oscillatory mode of the system when norelative transverse motion is allowed. It is noted that the system has no damping when7 approaches infinity. Since the system obviously has no damping when T is zero, theselection of the value of 7 to maximize system damping is critical.

Figure 4 contains root locus plots for equation (6) with respect to a set of nondi-mensional variables,

s* s -= Re* + i Im*WH

T* = TWH (9)

*0o - Co

w H

This relation allows a single figure representation, and also from practical considerationscH is of the order one. Only the upper left quadrant of the s* plane is presented, the

lower quadrant being a mirror image. Values of w* = 1.0, 1.5, and 2.0 and 7* vary-ing from zero to infinity were used to generate figure 4.

For no damping (7* = 0), the two positive roots of equation (6) are

1/2s* =ico*I +lI C* ( 002 + 4 ) 1 / 2 10/

1,2 =4+12 4 (10)

If the damping T* is increased, the root loci are seen to be approximately semi-circular, the positive limiting roots defined by equation (8) for 7* large being

s* = 0, 0, i (11)

This situation corresponds very nearly to that for a spin-stabilized, dual-spin-stabilized, or momentum-wheel-stabilized spacecraft and points out a major advantage ofan AMCD "passively" stabilized spacecraft, namely, the introduction of system damping.The maximum value for this damping for the "system" mode (the root which approaches

14

5

= 2.0

3

Im*

= 1.5

2

= 1.0

T* =0.4 Design 1point

( =2.8) 2.01 5

/ -

T= 1.6 0.8 10

1.6

0.8

-3 -2 -10

Re*

Figure 4.- Root locus of the basic "passive" system. EH =I = 0.

15

s* = i or s = iwH) which defines the "coning" motion associated with spin-stabilized

spacecraft can be approximated by assuming the root locus to be semicircular. This

assumption will yield

1- s(12)

(Psyst)max 1 (12)s

Although equation (12) predicts lower damping than that seen in figure 4, it is sufficient

to illustrate trends. Figure 5 is a plot of (Psyst)max determined from equation (12)

for selected values of Woo and wH. As can be seen from the figure, increasing the

AMCD momentum (increasing WH) or decreasing the bearing stiffness (decreasing woo)

will allow higher maximum system damping.

1.0

.9

.8

.7

.6

(Psyst 5

max

.3

.3H= 2 rad/sec

.2 1

.5.1

.25

0

Um, rad/sec

Figure 5.- System damping ratios.

From figure 5 it is seen that values of (Psyst)max could theoretically be as high

as one. However, from hardware considerations, there is a practical limit to the lead

T which must be taken into account in any specific design. An optimum selection of

system damping has been accomplished which minimizes an integral absolute spacecraft

pointing error criteria following an initial spacecraft offset. The value for the dimen-

sionless variable -* based on these simulation runs is

T* = 2.8

16

for a selected value of w* of 1.0. This value of T* was independent of WH for the

values of wcH selected which varied from WH = 0.25 to wH = 1.0. This value is

circled in figure 4 and can be used as a basis for an initial design of an AMCD "passive"

system, provided EI and EH are negligible.

Spacecraft Rotation Effects

For some applications nonnegligible spacecraft rotation may be a requirement

(EH t 0) and would occur, for example, for missions where Earth or satellite tracking was

necessary. Figure 6 contains root locus plots for increasing T* for values of eH = 0,0.25, and 0.5; w*= 1.0; R =2; and EI =0. Since the variables plotted are normalized

with respect to WH, the plots can be considered to represent a system with fixed total

momentum. Therefore, for a given vehicle, transferring momentum from the AMCD to

the spacecraft for the case where the spacecraft and AMCD rotate in similar directions

(EH > 0) can be represented by loci of roots of constant T* and increasing EH. The"system" root exhibits decreased damping resulting from the decrease in AMCD momen-

tum and this effect would require modification to these design criteria (r* = 2.8).

Figure 7 contains root locus plots for increasing T* for values of EH = 0, -0.1,

and -0.15; w* = 1; R = 3; and EI = 0. Again, a given vehicle can be represented by loci

of roots of fixed T* as eH increases negatively. The system root exhibits increased

damping, resulting from increased AMCD momentum.

AMCD Inertia Effects

The AMCD rim has a basic precessional mode with an uncoupled frequency (for the

thin rim assumption, Iaz = 2Ia) of

Ha2p = Ha = 24

a

Since the rim rotation rate 4 is large relative to other system frequencies, the

effect of the rim precessional mode on the system roots has been found to be negligible;

for example, in the generation of figures 4 to 7, no noticeable effect occurs when antici-

pated values of EI are included, and roots are found by use of equation (4). However, an

additional complex root is found which represents rim precession, and which requires ade-

quate magnetic bearing damping (proper selection of T*). Figure 8 contains plots of the

damping ratio for this rim mode as a function of bearing lead r* for values of E H = -0.5

to E H = 1.0 and for a value of E, = 0.0025 which results when nominal parameters are

17

2.5

20

1.5

*E

H= 0 ImT .4T = 1.6

-1.5 0 -.5 0

-I. 5 -I. 0 -.50

Re*Figure 6.- Effects of positive spacecraft rotation. = 0; w* = 1; R = 2.

18

18

2.0

1.5

T= .4

Im

1.0O

T =. 8/

-- T15 = 3.2

EHH0

EH-0 T 1.6

T -.15='8 T.4

T =3. 2

_ I l f II IIIi '

-1.0 -. 5 0

Re

Figure 7.- Effects of negative spacecraft rotation. E = 0; w* = 1; R = 2.

ORIGINAL PAGE IS 19OF POOR QUA

1.0 ' 1.0

12 3 4 56

Figure 8.- Rim damping ratios. EI = 0.0025.

selected. Values of the damped natural frequency of this mode did not vary significantly

from the uncoupled precessional frequency %p.

Simulation Results and Discussion

The nonlinear equations of motion for a passive AMCD-spacecraft configuration,which are derived in appendix C, were used to provide the time histories presented in

this section. The spacecraft that was simulated does not represent a particular vehicle

but should be representative of a typical Earth observation satellite. The configuration

is basically a right circular cylinder with a diameter of 1.52 meters (5 feet), a length of

2.5 meters (8.2 feet), and a mass of 1175 kilograms (80.5 slugs). The transverse iner-

tias are 680 kg-m 2 (500 slug-ft 2 ). An experiment-induced disturbance was used toillustrate the short-term dynamics of the system. The disturbances, shown in figure 9,represent the operation of protective doors and were taken from reference 7 (p. IX-17).

The AMCD parameters were derived by assuming a thin-rim configuration with a diameter

of 1.52 meters (5 feet) and a tip speed of 305 m/sec (1000 ft/sec). The parameters forthe magnetic bearings (that is, stiffness and rate gain) were derived from the information

20

presented in figure 4 for wH and w* of 1. From equations (5), since EI and EH

approach 0, the momentum Ha of the AMCD is equal to the transverse inertia Is of

the spacecraft. From equations (9) (with WH = 1), 7* = 7 and w* = woo. By using the

expression for wco2 in equations (5), the magnetic bearing stiffness K2 was found to be

7.78 N/cm (4.44 lb/in.). The T* for optimum system damping, as indicated in figure 4,

is 2.8. For purposes of illustration, one value of T* above and one value below the T*

for optimum damping were included in the simulation. These values were 6.4 and 0.8,

respectively.

Z

Z 2.76 - -

0.82

0.15 0.15 0.20

0.20 Time, see Time, sec-0.92

-2.46 F

Figure 9.- Experiment disturbance torques.

The results of the simulation runs are presented in figure s 10 to 14. Figures 10

to 12 represent a system with T* = 2.8. Figures 10 and 11 show system pointing accu-

racy and damping characteristics. The system damping is very good, system motion

being essentially damped out in approximately 3 cycles. Figure 12 is presented for the

purpose of illustrating practical air gap requirements. It represents the motion of the

rim center point within particular axial magnetic bearing segments so that the total peak-

to-peak motion would represent the total bearing gap required to allow uninterrupted

motion of the rim. The minimum bearing gap for the case considered is approximately

0.0012 cm (0.00047 in.). The bearing gaps would be made much larger than this in prac-

tice in order to ease machining tolerances. Figures 13 and 14 show system pointing

accuracy and damping characteristics for -* of 0.8 and 6.4, respectively.

DESCRIPTION OF OTHER POTENTIAL USES OF AN AMCD

An "Active" AMCD-Spacecraft System

The use of an AMCD as an active device involves the generation of spacecraft

torques in direct response to sensed spacecraft pointing errors. The torques would begenerated by energizing the AMCD magnetic bearing flux coils in response to desired

21

6

Oa

4

2

0

00

-2

-6L JL L UI I i IIIJ1 1 11 1 tili ! 11 1 11 1 1 111 10 5 10 15 20 25 30

Time, sec

Figure 10.- Response of AMCD to disturbance input with

* = 1 and T*= 2.8.

6

4

/s6

2

0

0

-2

-60 5 10 15 20 25 30

Time, sec

Figure 11.- Response of spacecraft to disturbance input withw = 1 and T* = 2.8.

22

15 X 10-4

10-

5 Bearings

13

E 0

-5

-10 -

0 5 10 15 20 25 30

Time, sec

Figure 12.- AMCD rim motion in magnetic bearings with

w* = 1 and T* 2.8.

6

4 -

S- 2

-2

Time, sec

Figure 13.- Response of spacecraft to disturbance input with

w* = 1 and r* = 0.8.

23

6

4

2

0 0

o -2

-4

-60 5 10 15 20 25 30

Time, sec

Figure 14.- Response of spacecraft to disturbance input with

w* = 1 and T* *=6.4.

control torques based on spacecraft sensor outputs. The spacecraft is reoriented by

producing torques against the AMCD rim momentum.

Two possible configurations of an active system are shown in figure 15. The spin-

axis-pointing configuration of figure 15(a) allows both spacecraft pitch and yaw axes to

be controlled by the magnetic bearing/torquers. However, if the target were the Earth

or Sun, this configuration would require application of an external torque to precess the

AMCD momentum vector at the appropriate rate. This configuration might be practical

for solar viewing from Earth orbit (10 per day precession) by producing torques against

the Earth's magnetic field with the use of conventional flux coils. Figure 15(b) illustrates

an alternative configuration in which the AMCD momentum vector is alined perpendicular

to the orbit plane for Earth viewing and perpendicular to the ecliptic plane for solar

viewing. This configuration allows complete spacecraft reorientation about the AMCD

spin axis with no requirement for direct external torques, orbit regression being

neglected. The second configuration does require that pointing control torques be gen-

erated about the spin axis; this was accomplished by using the spin motor with an inherent

increase in required power. The pointing field of view in both cases is limited at least

partially by the magnetic bearing gap and thus an additional mechanism is required for

spacecraft maneuvering.

24

Active magnetic bearing/torquer. 1

AMCD rim

Spacecraft

HAMC D

(a) Spin axis pointing.

HAMCD

(b) Spin axis perpendicular to pointing axis.

Figure 15.- Active AMCD-spacecraft system.

AMCD-Spacecraft Maneuvering

Large angle spacecraft reorientation or maneuvering can be accomplished in avariety of ways by using one or more AMCDs; however, only the two ways which appearto be most practical for large radius AMCDs are discussed. In figure 16(a), a single-gimbal, single AMCD for arbitrary orientation of the pointing axis of a spacecraft isshown. By making an analogy with Earth-based telescopes, the azimuth angle a isgenerated by AMCD spin-axis control, and the elevation angle /3 is generated by pro-ducing torques against the AMCD momentum on the spacecraft by using a gimbaltorquer(s). By limiting the spacecraft momentum during the maneuver to a small frac-tion of the AMCD momentum, one can see that the AMCD would remain nearly fixed, andthe spacecraft would move in a fashion similar to that of an Earth-based telescope. Whenthe spacecraft pointing axis is appropriately alined with the new target, the gimbal torquerwould be locked in place and fine pointing would be accomplished by using the active tech-nique discussed previously.

25

Extended arm HAMCD

AMCD rim

AMCD gimbal Magnetic bearing torquer

a Gimbaltorquer

(a) Single-gimbal, single-AMCD maneuver configuration.

H

H2

(b) Counterrotating AMCD pair configuration.

Figure 16.- Two AMCD-spacecraft maneuver configurations.

In figure 16(b) a second configuration is shown where two counterrotating AMCDs

are shown mounted directly to a spacecraft. By reorienting the two rims within their

respective magnetic bearing gaps or by using a limited gimbaling of one AMCD and'also

spinning up or despinning the rims, some momentum with appropriate direction can be

imparted to the spacecraft. This momentum represents a maneuvering spacecraft with

termination of the maneuver begun at a desired orientation. If all initial (prior to the

maneuver) nonzero momentum can be accommodated by the two AMCDs at the new orien-

tation, all spacecraft rates can be nulled. Thus, arbitrary maneuvers can be performed

with this system if given sufficient magnetic bearing gap or gimbal freedom and spin

speed range. Once the spacecraft has reached a new orientation, one or both of the

AMCDs can be used in an active or passive mode for spacecraft pointing or stabilization.

This configuration can also be used for combined power storage and control by maintain-

26

ing a net momentum difference between H 1 and H2 . The net momentum vector can

stabilize the spacecraft in a passive manner as discussed previously, and power can be

put into and taken from the system by spinning the two rims up or down while maintain-

ing a fixed difference in rotational speed.

Additional Applications

The AMCD can be used as a vernier experiment pointing system by mounting the

experiment to the rim and the magnetic bearing to the spacecraft (or vice versa) and

providing vernier pointing torques with the magnetic bearings by using fine pointing

information from the experiment package. The rim does not spin at high speed but the

spin motor is required for roll orientation and control. The magnetic bearings produce

torques against the spacecraft inertia and stabilization system rather than against the

rim momentum.

Although this report has emphasized the large radius applications of the AMCD,

the device may well be appropriate as the spin assembly for any conventional momentum

storage device, such as a reaction wheel, control moment gyroscope, or the gyroscopic

sensing systems found in aerospace vehicles.

CONCLUDING REMARKS

A device with potential broad application to spacecraft control and other momentum

storage applications has been defined and considerations of its configuration and use dis-

cussed. Examination of the device as a momentum storage unit and system indicates the

following advantages:

1. The rotating element approaches a thin rim which is the optimum shape for a

given stress-limited material when maximizing momentum for a given mass of material

and for a given maximum radius.

2. The thin rim allows a unidirectional filament layup of composites; thus, the

maximum usage of these high strength-weight materials is allowed.

3. The configuration allows, where possible and desirable, the use of a large rim

diameter with the inherent additional increase in momentum-mass ratio.

4. The noncontacting magnetic bearings and drive motor eliminate all mechanical

friction and wear and should yield a device reliability equal to that of the solid-state

electronic circuits.

5. The isolation of the rotating rim from the spacecraft affords an effective control

over transmittal of rim vibration to the spacecraft when active magnetic bearings with no

permanent magnetism are used. This is true because any forces transmitted to the

27

spacecraft would be the result of sensed rim motion and subsequent flux coil energizing,

and would thus allow appropriate notch filtering in the electronics at spin and vibrational

frequencies.

6. The magnetic bearings also provide the capability for directly producing torques

on the spacecraft with no mechanical or electrical breakout torques involved.

7. For the "passive" mode of spacecraft control (when compared with single-spin,dual-spin, or gyrostat control), much improved precessional damping can be shown

theoretically.

8. For the "passive" mode of spacecraft control, smaller attitude errors caused

by environmental torques will result from the higher momentum allowed by the AMCD

for a given momentum storage weight.

9. For the "active" mode of spacecraft control, extreme precision fine pointing is

projected since extremely low spacecraft control torques can be easily generated with

the magnetic bearings used as spacecraft torquers driven by spacecraft attitude sensors.

Finally, it is noted that the AMCD can be used as the spin assembly for any of the

conventional momentum storage control devices (such as reaction wheels or control

moment gyroscopes) with the unit characteristics of the AMCD offering the potential for

significant improvements.

Langley Research Center,

National Aeronautics and Space Administration,

Hampton, Va., December 19, 1974.

28

APPENDIX A

FLYWHEEL MOMENTUM DENSITY

By considering the constraint on flywheel speed to be determined by allowable

working stress, the specific energy of a flywheel can be determined from the relation-

ship (ref. 2)

TR K (Al)

m p

where TR is the stored kinetic energy; m, the mass; Ks, a dimensionless flywheel

shape factor; a, the design working stress; and p, the material density.

This relationship (eq. (Al)) can be modified to yield the specific momentum of a

flywheel by noting that

T - HwR 2(A2)

H = mk2,

where H is the angular momentum; w, the angular velocity; and k, the radius of gyra-

tion of the flywheel.

Substitution of equations (A2) into equation (Al) yields

H k(2Ks 1/2 (A3)

Shape factors for homogeneous materials are given in reference 2 for nine different

flywheel geometries and vary from 0.305 for a flat pierced disk to a theoretical limit of

1.000 for a constant-stress disk with an outer diameter approaching infinity. The shape

factor for a thin rim is 0.500. A factor of 0.85 represents a practical limit.

29

APPENDIX B

AMCD STIFFNESS EFFECTS

The equation of motion for AMCD rim vibrations in the spin plane is assumed to be

that of a rod under tension with rim thickness negligible when compared with rim radius.

The equation, from reference 8, is

orA a2 4 YI = pA 2 (B1)as 2 as 4 at 2

where a is the tangential stress; A, the cross-sectional area; Y, Young's modulus;

I, the area moment of inertia; p, the density of the rim; 71, the amplitude of in-plane

vibration; and s, the distance along the rim. The frequency of the fundamental mode

can be found from the following solution to equation (Bl):

S= 1o sin (2 -) sin wft (B2)

where ro is an arbitrary constant. Also it is noted that a = pr 2w 2 for a thin rim,where w is the spin frequency.

Substituting equation (B2) into equation (B1) and simplifying yields the relationship

Wf =V(2) 2 + 2,o (B3)

where

Wf, 0 = (B4)

From equation (B3) it can be seen that the vibration frequency of is always at

least twice as great as the spin frequency w. This formulation neglects rim inexten-

sibility for the sake of simplicity.

30

APPENDIX C

DERIVATION OF "PASSIVE" AMCD-SPACECRAFT EQUATIONS OF MOTION

The derivation begins by considering Euler's equations of motion in matrix nota-

tion for an arbitrary rigid body written with respect to a set of axes fixed to the body,

Tb = IbCOb + wbHb (C1)

where. Tb are the external torques; Ib, the inertia matrix; wo, the body rates of rota-

tion written as a cross product matrix; and Hb, the angular momentum. This body

represents the rotating rim of the AMCD.

Since the magnetic bearings which produce the torques Tb are fixed to the space-

craft, it is required to transform equation (CI) to a second arbitrarily oriented axis sys-

tem. This transformation Eab is defined by the equation

Va = Eabvb (C2)

where va and vb are arbitrary vectors.

The transformation of equation (Cl) begins by multiplying both sides by Eab and

by substituting, as indicated, the unity matrix taken as Ea5Eab. This procedure yields

E T = E IE-1, + E -1E H (C3)

EabTb = EabIbEaEab ab b + EabbEababHb (C3)

Equation (C3) can be simplified by defining

Ta = EabTb

Ia = EabIbEb 1

Ha = EabHb (C4)

w= Eab -bE-1 (matrix)

Wa = EabWb (vector)

By differentiating the last of equations (C4), it is seen that

d a = Eabd b + Eabb(C 5)

Eab(b = a - Eabb (

Substituting equations (C4) and (C5) into equation (C3) yields

Ta = Iac a + waHa - IaEabEalWa (C6)

31

APPENDIX C - Continued

By assuming the a-coordinate system and the b-coordinate system to have the same

Z-axis, with a rotation of about the Z-axis of the b-coordinate system relative to the

a-coordinate system,

cos -sin 0

S in P cos 01 (C7)

0 0 1

Equation (C6) can now be expanded and, by assuming AMCD rim symmetry

(Ibx = Iby = Ia), becomes

Tax = Ia Wax + (Iaz - Ia)waywaz + 4Iaway

Tay = Ia~cay + (Ia - Iaz)waxwaz - iIaWax (C8)

Taz = Iazcaz

Equations (C8) can now be modified to separate the effect of the large AMCD rim

spin velocity by introducing the variable waz which will represent the angular velocity

of the a-coordinate system about the Z-axis where

waz = w~z + i (C9)

Equations (C8) now become

Tax = IaCax + (Iaz - Ia)WaywCz + Iaz~ ay

Tay = Iac6ay + (Ia - Iaz)wCzwax - azax (C10)

Taz = Iazcaz + Iaz*I

These equations are the equations of motion of the AMCD rim written in the a-system.

A second set of Euler's equations for the spacecraft is now introduced, where the

algebra of the transformation of axes is omitted and where the AMCD and spacecraft

centers of mass are assumed to be coincident,

Gx + Tsx = Is sx + (Isz - Is)Wsysz + Isz Wsy

Gy + Tsy = IsCosy + (Is - Iaz)wszwsx - Isz 6 wsx (C11)

Gz + Tsz = isz sCz + Isz 6

where G is an external disturbance torque, spacecraft symmetry is assumed

(Isx = Isy = Is), and Wsz = wCz + 8. These equations are written in the s-system.

32

APPENDIX C - Continued

The rates waz and wCz are set equal to zero and the a- and s-coordinate axes

are assumed to be nearly coincident except for small transverse relative rotations. This

procedure allows a simplification in the calculation of the interaction torques by intro-

ducing a transformation matrix from the s-coordinate system to the a-coordinate system

as follows:

Eas = EaiEis = E2(0a) El( a)[E2 (0s) El(Ps)] -1 = E 2(0a) El(ca) E1 -l(os) E 2 - 1 (0s)

= E2(Oa) El(ca) El(-Os) E 2 (-s) = E2(0a) El(ba - Os) E 2(-Bs) (C12)

where the subscript i is introduced to represent an inertial reference and the notation

Ej (), j = 1 and 2, refers to a transformation from one coordinate system to another

which has been rotated through an angle denoted by ( ) about an axis j. The Euler

angles chosen to represent the positions of the a- and s-coordinate axes with respect to

an inertial axis set are 0 and 0 with a 1,2 rotation sequence (x,y) selected.

By expanding the matrices in equation (C12) for the case when ca - Os, Oa,and Os are small, the following equation is found to hold:

Eas - E2(Oa - Os) El(ca - Os) - El(Oa - Os) E 2 (Oa - Os) (C13)

Because of the small angles (Oa - 0s) and (Oa - Os) , Euler's theorem on rotations can be

used to show that

> = [(a - s)2+ (a - s)2] 1 / 2

sin i = a - 0 s

cos a s

di (a - 6s) cos i - (Oa - 0s) sin i

dt

where the second set of Euler angles X and i is shown in figure 17.

The torques on the AMCD rim resulting from "passive" or centering-only magnetic

bearing forces F 1 , F 2 , and F 3 (see fig. 17) can be expressed as a function of the rota-

tion X and the angle y and the linear magnetic bearing gains Kk and Kj as follows.

The bearing forces acting on the rim are defined by

F 1 = (KpX + K i)r sin y + KiXri cos y

F 2 = -(KjX + Ki)r sin (600 + y) - KlXr' cos (600 + y) (C14)

F 3 = (KX + Ki )r sin (600 - y) - K Xr cos (600 - y)

33

APPENDIX C - Continued

ZS

Axis of V

F3 1200 F2

X YsI

FI T,X,iXsf

Figure 17.- Magnetic bearing forces and

resultant torque.

where y is shown in figure 17. The sum of F 1 , F 2 , and F 3 is zero. The forces

are assumed to act perpendicular to the rim.

The torque is the resultant of the individual torques produced by the bearing forces

T = [-F 1 sin y + F 2 sin (600 + y)- F 3 sin (600 - y)] r

Tp = [-F 1 cos y + F 2 COS (600 + y) + F 3 Cos (600 - y)]r

Substituting equations (C14) into equations (C15) and simplifying yields

K-i)(C16)

Tp =-K X

where

K x = 1.5r 2 K((C17)

K = 1.5r 2 K J

Thus, for the case of 1200 spacing, the net bearing torque is not dependent on the position

of the magnetic bearing segments relative to the rim (the torque is independent of y) and

therefore the three linear bearings may be treated as two rotational bearings. The bear-

ing torques can now be written as

34

APPENDIX C - Concluded

Tax = -KX(a - Os)- Ki - s)- KS (Oa - s)(C18)

Tay = -Kx(Oa - Os)- KX(4a- 6s) + K@6( a - ,s)

The spin motor torque can be used to control either spacecraft Z-axis attitude or

attitude' rate as well as to counteract rim drag torque (hysteresis and eddy current

losses). In this derivation the component of motor torque which is greater than the

drag torque is of interest and

Taz = Tc - Td (C19)

where the subscripts c and d refer to control and drag.

Note that

Ts = -EsaTa (C20)

Thus, equations (C10) to (C12) and (C17) to (C20) represent the nonlinear AMCD-

spacecraft "passive" case equations of motion.

These equations 6an be linearized by assuming small angular motions except about

the spin axis, and for the case where Tc = Td and G = 0, they become

Ia a = -KX(Oa - s) - Ki(a - Ps) - HazOa - Ki6(Oa - 0s)

Ia0 a = -KX(Oa - s) - Kk(6a. - 5s) + Hazpa + KX6(Oa - Os)(C21)

Is s = KX(Oa - Ps) + Kk('a - 's) - Hsz0s + K6(0Oa - Os)

Is0s = KX(Oa - Os) + K (a - 5s) + Hszs, - K 6 (0a - OPs)

where

Hsz = Isz5

Haz = Iaz

The linearized equations for the Z-axis represent simple single-axis reaction-wheel

dynamics and are not considered.

35

REFERENCES

1. Notti, J. E.; Cormack, A., III; and Schmill, W. C.: Integrated Power/Attitude Control

System (IPACS) Study. Vol. I - Feasibility Studies. NASA CR-2383, 1974.

2. Lawsion, Louis J.: Design and Testing of igh Eerg Dsy ywee r Dpplica-

tion to Flywheel/Heat Engine Hybrid Vehicle Drives. 1971 Intersociety Energy

Conversion Engineering Conference, Soc. Automot. Eng., Inc., c.1971,

pp. 1142-1150.

3. Jacot, A. Dean; and Emsley, William W.: Assessment of Fine Stabilization Problems

for the LST. AIAA Paper No. 73-881, Aug. 1973.

4. Anon.: Proceedings of the Symposium on Attitude Stabilization and Control of Dual-

Spin Spacecraft 1-2 August 1967. Rep. No. SAMSO-TR-68-191, U.S. Air Force,

Nov. 1967. (Available from DDC as AD 670 154.)

5. Flatley, Thomas W.: Equilibrium States for a Class of Dual-Spin Spacecraft. NASA

TR R-362, 1971.

6. Mingori, D. Lewis: Stability of Linear Systems With Constraint Damping and Inte-

grals of the Motion. Astronaut. Acta, vol. 16, no. 5, Nov. 1971, pp. 251-258.

7. Anon.: Research and Applications Modules (RAM) Phase B Study. Rep.

No. GDCA-DDA 72-007 (Contract NAS 8-27539), General Dynamics Corp.,May 12, 1972. (Available as NASA CR-123785.)

8. Anderson, Willard W.: On Lateral Cable Oscillations of Cable-Connected Space

Stations. NASA TN D-5107, 1969.

36 NASA-Langley, 1975 L-9901